MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sb4e Structured version   Unicode version

Theorem sb4e 1949
Description: One direction of a simplified definition of substitution that unlike sb4 2093 doesn't require a distinctor antecedent. (Contributed by NM, 2-Feb-2007.)
Assertion
Ref Expression
sb4e  |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  E. y ph ) )

Proof of Theorem sb4e
StepHypRef Expression
1 sb1 1662 . 2  |-  ( [ y  /  x ] ph  ->  E. x ( x  =  y  /\  ph ) )
2 equs5e 1910 . 2  |-  ( E. x ( x  =  y  /\  ph )  ->  A. x ( x  =  y  ->  E. y ph ) )
31, 2syl 16 1  |-  ( [ y  /  x ] ph  ->  A. x ( x  =  y  ->  E. y ph ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 359   A.wal 1549   E.wex 1550   [wsb 1658
This theorem is referenced by:  hbsb2e  2098  hbsb2eNEW7  29543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-11 1761
This theorem depends on definitions:  df-bi 178  df-an 361  df-ex 1551  df-nf 1554  df-sb 1659
  Copyright terms: Public domain W3C validator